Heat 1 tbsp olive oil in large nonstick skillet over medium-high heat. Add onion, just 1 tsp thyme, and a pinch of salt and pepper and cook, stirring occasionally, until softened and lightly browned, 8 to 10 minutes. Off heat, stir in just half the vinegar. (TIP: You can cook onion for longer for more caramelization and deeper flavor.)

Toast sourdough. Whisk together remaining vinegar, just half the Dijon, 1 tbsp olive oil, and a pinch of salt and pepper in medium bowl. Add arugula and toss to coat.

Arrange toasted sourdough on clean work surface. Spread herb garlic cheese on 2 slices and top with French onion filling and dressed arugula. Spread remaining Dijon on remaining sourdough slices and close sandwiches. Cut each in half and divide between serving plates.